Question

Which value, to the nearest tenth, is the smallest solution of f(x) = g(x) if f(x) = ln(x + 5) – 1 and g(x) = x3 – 2x + 1?

Answer 1

Answer:

There are two possibilities of determine which solution is the smallest one:

(i) The smallest solution is the value of $x$ has the smallest distance with respect to origin: $(x_{2},y_{2}) = (0.18,0)$

(ii) The smallest solution is the most negative value: $(x_{1},y_{1}) = (-1.577,0)$

Step-by-step explanation:

Let $f(x) = g(x)$, $f(x) = \ln (x+5)-1$ and $g(x) = x^{3}-2\cdot x + 1$, then we have the following implicit equation:

$\ln (x+5)-x^{3}+2\cdot x -2 = 0$ (1)

This equation cannot be solved by analytical means. There are two different strategies to solve this expression: (i) Numerical methods, (ii) Graphical methods. We decide to use the second method:

Let suppose that we use this function $h(x) = \ln (x+5) -x^{3}+2\cdot x - 2$. By the help of a graphing tool, we find the smallest solution such that $h(x) = 0$. There are three possible solutions: $(x_{1},y_{1}) = (-1.577,0)$, $(x_{2},y_{2}) = (0.18,0)$, $(x_{3}, y_{3}) = (1.376, 0)$

There are two possibilities of determine which solution is the smallest one:

(i) The smallest solution is the value of $x$ has the smallest distance with respect to origin: $(x_{2},y_{2}) = (0.18,0)$

(ii) The smallest solution is the most negative value: $(x_{1},y_{1}) = (-1.577,0)$